Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Non-gaussian geometrical measures in redshiftspace
Dmitry Pogosyan
Physics DepartmentUniversity of Alberta
April 10, 2019
with: S. Codis, C. Pichon, F. Bernardeau, T. Matsubara
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Density in redshift space
ρs (X, v ) =
∫
d z ρ(x)1
p
2πβT (x)exp
−(v − vc(x)−u (x))2
2βT (x)
Cold flow: βT → 0 (effectivly never achieved due to finite redshift rezolution)
ρs (X, v ) =
∫
d z ρ(x)δD (v − vc(x)−u (x))
ISM/turbulence studiesNo detailed predictions from underlyingtheory: mode content or relationbetween velocities and density are notknown apriori.Lesson: slice and dice PPV cube,synthetically varying some controlparameters to disentangle effects.Example: measuremets in velocitychannels of variable width.
CosmologyGood predictive understanding oftheory that lead to PPZ.Natural approach: try to fit fullpredictions of power and higher orderspectra.Still valuable - measure some integralsof spectra that take us straight tocosmological info. Having controlparameters to vary in your analysishelps.
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Geometrical measures for random fields
• Properties of ν= c o n s t isocontours - Minkowski functionals(genus/Euler characteristics χ(ν) , length of isocontours, pencilbeam isocontour crossing statistics, ND (ν))
• Statistics of extrema - peaks, minima, saddles• Skeleton of the structure, its statistics (length, curvature)• . . .
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Geometrical measures for random fields
Starting pointLet us think about such properties of a random field ρ as Eulercharacteristic (genus), density of maxima, length of skeleton. Theircomputation reguire knowledge of the joint distribution
P (ρ,ρi ,ρi j , . . .)
of the field ρ and its first ρi , second ρi j (Hessian matrix) and perhapshigher derivatives, for instance
nma x (ν) =
∫
0≥λ1≥λ2≥...
P (ρ = ν,ρi ,ρi j )δ(ρi )|ρi j |dρi j
Usual approach is to deal with it in the Hessian eigenvalue space, sincethat’s where the boundary conditions are the simplest.
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Non-Gaussian expansion for geometrical statistics
• To treat non-Gaussianities the idea is to expand P (ρ,ρi ,ρi j ) intoorthogonal polinomials around the Gaussian approximation like
P (x ) =G (x )
1+∑
n ⟨x n ⟩c Hn (x )
• The trick to avoid difficulties is an appropriate choice of variables:• that are invariant wrt symmetries of the problem (isotropy)• that are polynomial in the field quanities (λ’s are no good)• that simplify the Gaussian limit, being as uncorrelated as possible
• Useful set is: I1, · · · , IN , q 2, ζ≡ ρ+γI11−γ2
where In are N polynomial rotation invariants of the Hessian matrix ρi j ,I1 = Trρi j , . . . , IN = det |ρi j |
(and I2 . . . IN−1 are built from the minors of orders 2 to N-1 )
• Actually, better to use more ’irreducible’ combinations Ji , in N D -space
J1 = I1 , Js≥2 = I s1 −
s∑
p=2
(−N )p C ps
(s −1)C pN
I s−p1 Ip
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Orhogonal polymonial expansion for 2D P (ρ, q 2, J1, J2)
Gaussian limit JPDF
G2D(ζ, q 2, J1, J2) dζdq 2dJ1dJ2 =1
2πe −
12 (ζ2+2q 2+J 2
1 +2J2)dζdq 2dJ1dJ2
serves as the weight for defining the expansion polynomials in
• ζ, J1 – ([−∞,∞], gaussian weight) – Hermite
• q 2, J2 – ([0,∞], exponential weight) – Laguerre.
P2D(ζ, q 2, J1, J2) =G2D ×
1+
∞∑
n=3
i+2 j+k+2l=n∑
i , j ,k ,l=0
(−1) j+l
i ! j ! k ! l !
¬
ζi q 2 jJ1
k J2l¶
GCHi (ζ)L j
q 2
Hk (J1)L l (J2)
This is an expansion to all orders in powers of the field n .Expansion coefficients can be predicted by pertrubation theories
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Euler characteristic (genus) as a function of threshold
General expression in N D
χ(ν)2= (−1)N
∫ ∞
ν
d x
∫
d q 2q N−1δND (q
2)
∫ N∏
s=1
d JsPND(. . .) IN
can be integrated to give “moment” expansion to all orders
χ(ν) =1
p2πR∗
exp
−ν2
2
×2
(2π)N /2
γp
N
N
HN−1(ν)+
+∞∑
n=3
N∑
s=0
γ−si+2 j=n−s∑
i , j=0
(−N ) j+s (N −2)!!L( N−2
2 )j (0)
i !(2 j +N −2)!!
¬
x i q 2 jIs
¶
G CHi+N−s−1(ν)
(cf first term: Matsubara 1994-2005)
with coefficients that can be predicted by perturbation theories
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
How non-Gaussianity develops, eq 2D Euler charachteristic
Excitation of Hermite modes of alternating parity
- 4 -2 0 2 4
-0.005
0.000
0.005
Ν
Χ
-4 -2 0 2 4
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
Ν
DΧ
Σ = 0.10
-4 -2 0 2 4
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
Ν
DΧ
Σ = 0.16
-4 -2 0 2 4
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
Ν
DΧ
Σ = 0.20
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
“Hermite spectroscopy”
From:
2563
n=-1
-3 -2 -1 1 2 3Νf
-8
-6
-4
-2
2
4
106 Χ3D, f
Theory for Ν
Theory for ΝfMeasurements
To:
H0 H1 H2 H3 H4 H5 H6
0.2
0.4
0.6
0.8
1.0
1.2
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Redshift space – anisotropic statistics
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Polymonial expansion for P3D in redshift space
Symmetry:Rotational around the line of sight (LOS along 3rd coordinate)
Variables:
linear(4) x , x3, x33, J1⊥ = x11+ x22
quadratic(3) q 2 = x 21 + x 2
2 , Q 2 = x 213+ x 2
23, J2⊥ = (x11− x22)2+4x 212
cubic(1) Υ =
x 213− x 2
23
(x11− x22)+4x12 x13 x23
Gaussian limit JPDF
G (x ,q 2⊥,x3 ζ,J2⊥,ξ,Q 2,Υ ) =
1
4π3p
Q 4 J2⊥−Υ 2e −
12 x 2−q 2
⊥−12 x 2
3−12ζ
2−J2⊥− 12ξ
2−Q 2
Uniformly distributed Υ ∈ [−Q 2p
J2⊥,+Q 2p
J2⊥] can be integrated over forMinkowski functionals, extrema . . .
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Euler characteristic in redshift space
χ2+1(ν) =e −ν
2/2
8π2
σ1‖σ21⊥
σ3H2(ν) +
∞∑
n=3
χ (n )2+1
with non-Gaussian corrections χ (n )2+1, given, to all orders, by
χ (n )2+1(ν) =σ2
2⊥σ2‖
σ21⊥σ1‖
∑
σn
(−1) j+m
2m i ! j ! m !Hi+2(ν)γ‖γ
2⊥
¬
x i q 2 j⊥ x 2m
3
¶
GC
−∑
σn−1
(−1) j+m
2m i ! j ! m !Hi+1(ν)
¬
x i q 2 j⊥ x 2m
3
γ2⊥x33+2γ⊥γ‖ J1⊥
¶
GC
+∑
σn−2
(−1) j+m
2m i ! j ! m !Hi (ν)
¬
x i q 2 j⊥ x 2m
3
2γ⊥(J1⊥x33−γ2Q 2) +γ‖(J2
1⊥− J2⊥)
¶
GC
−∑
σn−3
(−1) j+m
2m i ! j ! m !Hi−1(ν)
¬
x i q 2 j⊥ x 2m
3
x33(J2
1⊥− J2⊥)−2γ2
Q 2 J1⊥−Υ
¶
GC
with coefficients that can be predicted by perturbation theories
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Euler characteristics in redshift space
σ= 0.18, f = 1
2563
n=-1
-4 -2 2 4Ν
-10
-5
5
106 Χ3D
IsotropicAnisotropicMeasurements
2563
n=-1
-4 -2 2 4Ν
-3
-2
-1
1
2
3
106 DΧ3D
IsotropicAnisotropicMeasurements
χ0+13D =
σ1‖σ21⊥
σ3
e −ν2/2
8π2
H2(ν) +1
3!H5(ν)
x 3
+H3(ν)
x q 2⊥
+
x x 23
2
−H1(ν)γ⊥
J1⊥q 2⊥
+
J1⊥x 23
Important anisotropy measure βσ = 1− σ21⊥
2σ21‖≈ 4
5 f /b
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Slicing through redshift space
χ2D(ν,θ ) =e −ν
2/2
(2π)3/2σ2
1⊥2σ2
√
√
√ 1−βσ sin2 θ
1−βσ×
H1 (ν) +1
3!H4 (ν)
x 3
+H2(ν)
x q 2⊥
+1
2
cos2θ
1−βσ sin2 θ
x x 23
−
x q 2⊥
−1
γ⊥
q 2⊥ J1⊥
+1
2
cos2θ
1−βσ sin2 θ
x 23 J1⊥
−2
q 2⊥ J1⊥
+O (σ2)
N2(ν,θ ) =σ1⊥
2p
2σe −ν
2/2
1+βσcos2 θ
4
×
1+1
3!H3(ν)
x 3
+1
2H1(ν)
x q 2⊥
+1
2cos2 θ
1+βσ3+5 sin2 θ
8
x x 23
−
x q 2⊥
+O (σ2)
N1(ν,θ ) =σ1⊥p3πσ
e −ν2/2
√
√
√ 1−βσ sin2 θ
1−βσ×
1+1
3!H3(ν)
x 3
+1
2H1(ν)
x q 2⊥
+cos2θ
1−βσ sin2 θ
x x 23
−
x q 2⊥
+O (σ2)
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Invariance of Minkowski functionals
• Minkowski functionals ( but also extrema counts ) are invariant under any localmonotonic transformations x = f (y ) , if one adjust threshold accordingly,νx = f (νy ). Which is achived by choosing threshold that gives the same fillingfactor of the volume above.
• This means that the coefficients in front of every mode in Hermite expansion areinvariant separately. This can be shown perturbatively (?).
• Corollary I: combinations of cumulants in every Hermite mode are invariant wrt toany local monothonic bias. (Is it order by order in σ expansion ?)
• Corollary II: If the starting field y is Gaussian, these coefficients are zero (or sum to zero ?)
• Corollary IIa: In anisotropic case, angle dependent section of each coefficient should thenvanish separately. In particular
⟨x q 2⊥⟩= ⟨x x 2
3 ⟩ →⟨x∇⊥x ·∇⊥x ⟩⟨x∇‖ ·∇‖⟩
=σ2
1⊥σ2
1‖
• This holds for fN L models in leading non Gaussian model explicitly (despite even
transformation being local but non-monotonic in this case)
• But here we are talking about local transformation in redshift space.
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Fiducial experiment
• Cut through redshisift space at different angles. For instance, θ = 0 and θ =π/2and measure 2D geometrical statistics as the function of filling factor
• Do Hermite decomposition, determine Hn coefficients
• Use the ratio of main Gaussian lines to determine βσ = 1− σ21⊥
2σ1‖
H1
(χ2D (ν f ,π/2)
H1
χ2D (ν f , 0) =
Æ
1−βσ+O (σ2)H0
N2(ν f ,π/2)
H0
N2(ν f , 0) =
Æ
1−βσ+O (σ2)
Different statistics have different corrections O (σ2), which can be leveraged forcontrol
• Use angle variable ratio of a secondary to Gaussian line and βσ to determinenormalized cumulant combinations, and their anisotropy
H2
N2(ν f ,θ )
H0
N2(ν f ,θ ) → ⟨x q 2
⊥⟩ , ⟨x q 2⊥⟩− ⟨x x 2
3 ⟩
• Using theoretical insight ( PT ) relate βσ and higher order redshift cumulants toreal space σ, f , b . . . . To evaluate σ 3D statistics have most power.
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Using non-Gaussianity and redshift distortions of geometricalmeasures of the Cosmic Web to recover
β =Ω0.55/b
-1.0 -0.5 0.5 1.0Ν
0.82
0.84
0.86
0.88
0.90
N2,¦even N2,þ
evenΒ=0.8Β=0.9Β=1Β=1.1Β=1.2
Reconstructed β from angulardependence of Minkowskifunctionals in 2D slices
σ ( and D(z) )
-1.0 -0.5 0.5 1.0Ν
-0.3
-0.2
-0.1
0.1
N2,þoddNm
Σ=0.1Σ=0.15Σ=0.2Σ=0.25Σ=0.3
Reconstructed σ fromnon-Gaussian corrections ofMinkowski functionals
Redshift space non-Gaussian fields Anistropic non-Gaussian fields Summary
Conclusions
• Redshift space distortions are not just nuisance, but a source ofadditional information about cosmology of the Universe.
• Redshift space analysis is in principle capable of mining more informationthan real space analysis. For that one needs to leverage the anisotropicproperties of redshift space.
• Generalizing previous work, we have developed complete theory ofMinkowski functionals in the bulk and on slices of anisotropic and mildlynon-Gaussian fields.
• At mildly non-linear scales of cosmological structure formation,non-Gaussian and redshift effects can be, to some extend, disentangled.